Heuristic Policies for Stochastic Knapsack Problem with Time-Varying Random Demand

TL;DR

This paper analyzes the stochastic knapsack problem with time-varying demand, establishing properties of optimal policies and proposing heuristic switch-over policies that perform near-optimally in numerical tests.

Contribution

It extends the analysis of optimal policies to time-dependent demand distributions and introduces a new class of heuristic switch-over policies for the problem.

Findings

Monotonicity properties hold with time-dependent prices.

Bounds for value functions are developed for arbitrary demand sizes.

Switch-over policies perform close to optimal in experiments.

Abstract

In this paper, we consider the classic stochastic (dynamic) knapsack problem, a fundamental mathematical model in revenue management, with general time-varying random demand. Our main goal is to study the optimal policies, which can be obtained by solving the dynamic programming formulated for the problem, both qualitatively and quantitatively. It is well-known that when the demand size is fixed and the demand distribution is stationary over time, the value function of the dynamic programming exhibits extremely useful first and second order monotonicity properties, which lead to monotonicity properties of the optimal policies. In this paper, we are able to verify that these results still hold even in the case that the price distributions are time-dependent. When we further relax the demand size distribution assumptions and allow them to be arbitrary, for example in random batches, we develop a scheme for using value function of alternative unit demand systems to provide bounds to the value function. These results confirm some of the basic understandings of the stochastic knapsack problem. As a natural consequence, we propose a new class of heuristic policies, two-dimensional switch-over policies, and discuss numerical procedures for optimizing the parameters of the policies. Extensive numerical experiments demonstrate that these switch-over policies can provide performances that are in a close neighborhood of the optimum statistically.